Logarithmic scaling for fluctuations of a scalar concentration in wall turbulence

Within wall turbulence, there is a sublayer where the mean velocity and the variance of velocity fluctuations vary logarithmically with the height from the wall. This logarithmic scaling is also known for the mean concentration of a passive scalar. By using heat as such a scalar in a laboratory experiment of a turbulent boundary layer, the existence of the logarithmic scaling is shown here for the variance of fluctuations of the scalar concentration. It is reproduced by a model of energy-containing eddies that are attached to the wall.

Figure 1

Velocity statistics $U$, $\langle u^2\rangle$, $\langle w^2\rangle$, and $\langle -uw\rangle$ as a function of the height $z$. They have been obtained by heating the floor as ${\mathrm{\Theta }}_0-{\mathrm{\Theta }}_{\infty }=+3$ K ($▵$), by cooling the floor as $-3$ K ($▿$), or by neither heating nor cooling the floor ($\circ$). For the constant-flux sublayer of the last case, the straight lines denote Eqs. (1a) and (2a) with the parameter values in Table 1. The range of this sublayer is also shown. Although error bars of $1\sigma$ lie on all the data, none of them is discernible.

Figure 2

Temperature statistics $\mathrm{\Theta }-{\mathrm{\Theta }}_{\infty }$, $\langle {\theta }^2\rangle$, $\langle {\theta }^3\rangle /{\langle {\theta }^2\rangle }^{3/2}$, and $\langle {\theta }^4\rangle /{\langle {\theta }^2\rangle }^2$ as a function of the height $z$. They have been obtained by heating the floor as ${\mathrm{\Theta }}_0-{\mathrm{\Theta }}_{\infty }=+3$ K ($▴$) or by cooling the floor as $-3$ K ($▿$). For the constant-flux sublayer of the latter case, the straight lines denote Eqs. (1b) and (2b) with the parameter values in Table 2. The range of this sublayer is also shown. Although error bars of $1\sigma$ lie on all the data, only a few of them are discernible.

Figure 3

Schematics of attached eddies and of the moments $I_{\mathrm{\Theta }}(z/h_e)$, $I_{{(\mathrm{\Theta }+\theta )}^2}(z/h_e)$, and $I_{\theta w}(z/h_e)$ defined in Eqs. (5b), (8b), and (10b), where gray areas correspond to the undermost portions of the eddies.